Projective bundles

Baker, R. D.; Brown, J. M. N.; Ebert, G. L.; Fisher, J. Chris

doi:10.36045/bbms/1103408578

Cited by 15 publications

(14 citation statements)

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“…This observation is useful when counting special conics in this article. We also note that the set of q 2 + q + 1 (π, ℓ ∞ )-special conics of π form a circumscribed bundle of conics in the sense of [3].…”

Section: Special Conicsmentioning

confidence: 97%

The exterior splash in PG(6, q): carrier conics

Barwick

Jackson

2017

Advances in Geometry

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Let π be an order-q-subplane of PG(2, q 3 ) that is exterior to ℓ ∞ . The exterior splash of π is the set of q 2 + q + 1 points on ℓ ∞ that lie on an extended line of π. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry CG(3, q), and hyper-reguli of PG (5, q). In this article we use the Bruck-Bose representation in PG(6, q) to give a geometric characterisation of special conics of π in terms of the covers of the exterior splash of π. We also investigate properties of order-q-subplanes with a common exterior splash, and study the intersection of two exterior splashes.

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Section: Special Conicsmentioning

confidence: 97%

The exterior splash in PG(6, q): carrier conics

Barwick

Jackson

2017

Advances in Geometry

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“…Under the Veronese bijective correspondence between conics of PG(2, q) and points of PG(5, q), let us consider V as the Veronese surface arising from the lines of PG(2, q) counted twice and V ′ as the Veronese surface obtained from the conics of an inscribed projective bundle, i.e. a particular collection of non-degenerate conics of PG(2, q) that mutually intersect in exactly one point, see [1]. By using the classification of pencils of conics of PG(2, q), [15, Table 7.7], [17], it is possible to show that V, V ′ satisfy the hypothesis of the previous theorem.…”

Section: A Geometric Descriptionmentioning

confidence: 99%

Small complete caps in ${\rm PG}(4n + 1, q)$

Cossidente,

Csajbók,

Marino

et al. 2021

Preprint

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In this paper we prove the existence of a complete cap of PG(4n + 1, q) of size 2(q 2n+1 − 1)/(q − 1), for each prime power q > 2. It is obtained by projecting two disjoint Veronese varieties of PG(2n 2 + 3n, q) from a suitable (2n 2 − n − 2)-dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of PG(4n + 1, q) is essentially sharp.

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“…When q is odd, he showed the existence of three distinct types of projective bundles in PG(2, q), by identifying them with planes in PG(5, q). It was shown in [2] that perfect difference sets can also be used to describe these projective bundles. In fact, given a perfect difference set D ⊆ Z/(q 2 + q + 1)Z and its circular shifts corresponding to the set of lines of PG(2, q), the three bundles are represented in the following way.…”

Section: Mdpc Codes From Projective Bundlesmentioning

confidence: 99%

Moderate Density Parity-Check Codes from Projective Bundles

Jessica¹,

Mattheus²,

Neri³

et al. 2021

Preprint

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A new construction for moderate density parity-check (MDPC) codes using finite geometry is proposed. We design a parity-check matrix for this family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine minimum distance and dimension of these codes, showing that they have a natural quasi-cyclic structure. In addition, we analyze the error-correction performance within one round of a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance for this range of parameters.

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Projective bundles

Cited by 15 publications

References 0 publications

The exterior splash in PG(6, q): carrier conics

The exterior splash in PG(6, q): carrier conics

Small complete caps in ${\rm PG}(4n + 1, q)$

Moderate Density Parity-Check Codes from Projective Bundles

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